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National and Regional Contests
Russia Contests
All-Russian Olympiad
1980 All Soviet Union Mathematical Olympiad
287
287
Part of
1980 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 287 All Soviet Union MO 1980 |AM| + |AP| = a, area of ABCD < a^2 /2
Source:
7/19/2019
The points
M
M
M
and
P
P
P
are the midpoints of
[
B
C
]
[BC]
[
BC
]
and
[
C
D
]
[CD]
[
C
D
]
sides of a convex quadrangle
A
B
C
D
ABCD
A
BC
D
. It is known that
∣
A
M
∣
+
∣
A
P
∣
=
a
|AM| + |AP| = a
∣
A
M
∣
+
∣
A
P
∣
=
a
. Prove that
A
B
C
D
ABCD
A
BC
D
has area less than
a
2
2
\frac{a^2}{2}
2
a
2
.
geometry
areas
Locus
midpoints
convex